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Thread: Determinant, Change of basis

  1. #1
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    Determinant, Change of basis

    Hi!

    Let $\displaystyle V$ be a vector space, $\displaystyle \dim (V)=n$ and $\displaystyle \omega \in \mathrm{Alt}^n V,\ \omega \neq 0 $ an alternating form. Let $\displaystyle M$ be the change of basis matrix from the basis $\displaystyle a=(a_1,...,a_n)$ for $\displaystyle V$ to the basis$\displaystyle b=(b_1,...,b_n) $ for $\displaystyle V$.

    Proof that $\displaystyle \det M=\frac{\omega(a_1,...,a_n)}{\omega(b_1,...,b_n)}
    $

    I would like to use the Leibniz formula
    $\displaystyle \det M = \sum_{\sigma \in S_n} \mathrm{sign} (\sigma) \cdot M_{1 \sigma(1)}\cdot ... \cdot M_{n \sigma(n)} $

    Since $\displaystyle a=(a_1,...,a_n)$ and $\displaystyle b=(b_1,...,b_n)$ are bases for $\displaystyle V$, I can write

    $\displaystyle a_i = \sum_{j=1}^{n}M_{ij}b_j$
    and therefore

    $\displaystyle \frac{\omega(\sum_{j=1}^{n}M_{1j}b_j,..., \sum_{j=1}^{n}M_{nj}b_j)}{\omega(b_1,...,b_n)} $


    $\displaystyle =\frac{\omega( M_{11}b_1+...+M_{1n}b_n ,..., M_{n1}b_1+...+M_{nn}b_n )}{\omega(b_1,...,b_n)} $

    How should I continue? Can I write the last as $\displaystyle \frac{\sum_{j=1}^n M_{1j} \cdot ... \cdot M_{nj} \cdot \omega(b_1,...,b_n)}{\omega(b_1,...,b_n)} $ because $\displaystyle \omega$ is alternating?

    Thanks in advance!

    Bye,
    Lisa
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  2. #2
    MHF Contributor

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    Quote Originally Posted by lisa View Post
    Hi!

    Let $\displaystyle V$ be a vector space, $\displaystyle \dim (V)=n$ and $\displaystyle \omega \in \mathrm{Alt}^n V,\ \omega \neq 0 $ an alternating form. Let $\displaystyle M$ be the change of basis matrix from the basis $\displaystyle a=(a_1,...,a_n)$ for $\displaystyle V$ to the basis$\displaystyle b=(b_1,...,b_n) $ for $\displaystyle V$.

    Proof that $\displaystyle \det M=\frac{\omega(a_1,...,a_n)}{\omega(b_1,...,b_n)}
    $

    I would like to use the Leibniz formula
    $\displaystyle \det M = \sum_{\sigma \in S_n} \mathrm{sign} (\sigma) \cdot M_{1 \sigma(1)}\cdot ... \cdot M_{n \sigma(n)} $

    Since $\displaystyle a=(a_1,...,a_n)$ and $\displaystyle b=(b_1,...,b_n)$ are bases for $\displaystyle V$, I can write

    $\displaystyle a_i = \sum_{j=1}^{n}M_{ij}b_j$
    and therefore

    $\displaystyle \frac{\omega(\sum_{j=1}^{n}M_{1j}b_j,..., \sum_{j=1}^{n}M_{nj}b_j)}{\omega(b_1,...,b_n)} $


    $\displaystyle =\frac{\omega( M_{11}b_1+...+M_{1n}b_n ,..., M_{n1}b_1+...+M_{nn}b_n )}{\omega(b_1,...,b_n)} $

    How should I continue? Can I write the last as $\displaystyle \frac{\sum_{j=1}^n M_{1j} \cdot ... \cdot M_{nj} \cdot \omega(b_1,...,b_n)}{\omega(b_1,...,b_n)} $ because $\displaystyle \omega$ is alternating?

    Thanks in advance!

    Bye,
    Lisa
    the point here is that since $\displaystyle \omega$ is alternating we have $\displaystyle \omega(x_1, \cdots , x_n)=0$ if $\displaystyle x_i=x_j$ for some $\displaystyle i \neq j.$ so using multilinearity of $\displaystyle \omega$ to expand $\displaystyle \omega(\sum_{j=1}^{n}M_{1j}b_j,..., \sum_{j=1}^{n}M_{nj}b_j),$ all terms in which one of $\displaystyle b_j$ appears more than once will be zero. thus we'll be left with terms in the form $\displaystyle c_{\sigma} \omega(b_{\sigma(1)}, \cdots , b_{\sigma(n)}),$ where $\displaystyle \sigma \in S_n$ and $\displaystyle c_{\sigma}$ is in terms of $\displaystyle M_{ij}.$ again, using the fact that $\displaystyle \omega$ is alternating, we have

    $\displaystyle \omega(b_{\sigma(1)}, \cdots , b_{\sigma(n)})=\text{sgn}(\sigma) \cdot \omega(b_1, \cdots , b_n).$ the only thing you need to show now is that $\displaystyle c_{\sigma}=M_{1\sigma(1)} \cdot \cdots M_{n \sigma(n)}.$ (left for you!)
    Last edited by NonCommAlg; May 17th 2010 at 05:03 AM.
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